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21: Bibliography
  • D. W. Albrecht, E. L. Mansfield, and A. E. Milne (1996) Algorithms for special integrals of ordinary differential equations. J. Phys. A 29 (5), pp. 973–991.
  • G. D. Anderson, S.-L. Qiu, M. K. Vamanamurthy, and M. Vuorinen (2000) Generalized elliptic integrals and modular equations. Pacific J. Math. 192 (1), pp. 1–37.
  • F. M. Arscott (1964a) Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.
  • 22: Bibliography L
  • C. G. Lambe and D. R. Ward (1934) Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.
  • 23: 10.1 Special Notation
    For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
    24: 28.31 Equations of Whittaker–Hill and Ince
    For proofs and further integral equations see Urwin (1964, 1965). …
    25: 19.2 Definitions
    19.2.8_1 K ( k ) = 0 1 d t 1 t 2 1 ( 1 k 2 ) t 2 ,
    19.2.8_2 E ( k ) = 0 1 1 ( 1 k 2 ) t 2 1 t 2 d t ,
    19.2.11_5 el1 ( x , k c ) = 0 arctan x 1 cos 2 θ + k c 2 sin 2 θ d θ ,
    26: Bibliography B
  • A. P. Bassom, P. A. Clarkson, A. C. Hicks, and J. B. McLeod (1992) Integral equations and exact solutions for the fourth Painlevé equation. Proc. Roy. Soc. London Ser. A 437, pp. 1–24.
  • 27: 1.4 Calculus of One Variable
    1.4.23_1 α ( d ) α ( c ) = c d w ( x ) d x , [ c , d ] I .
    1.4.23_2 a b f ( x ) d α ( x ) = a b f ( x ) w ( x ) d x , f integrable with respect to d α .
    1.4.23_3 a b f ( x ) d α ( x ) = a b w ( x ) f ( x ) d x + n = 1 N w n f ( x n ) .
    1.4.34 𝒱 a , b ( f ) = a b | f ( x ) | d x ,
    28: 15.9 Relations to Other Functions
    15.9.25 E ( k ) = π 2 F ( 1 2 , 1 2 1 ; k 2 ) ,
    15.9.26 D ( k ) = π 4 F ( 1 2 , 3 2 2 ; k 2 ) .
    29: 7.18 Repeated Integrals of the Complementary Error Function
    7.18.4 d n d z n ( e z 2 erfc z ) = ( 1 ) n 2 n n ! e z 2 i n erfc ( z ) , n = 0 , 1 , 2 , .
    30: 36.15 Methods of Computation
    For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).